Second Moment Method on k-SAT: a General Framework

We give a general framework implementing the Second Moment Method on k-SAT and discuss the conditions making the Second Moment Method work in this framework. As applications, we make the Second Moment Method work on boolean solutions and implicants. We extend this to the distributional model of k-SAT.Comment: 35 page

The Scaling Window of the 2-SAT Transition

We consider the random 2-satisfiability problem, in which each instance is a formula that is the conjunction of m clauses of the form (x or y), chosen uniformly at random from among all 2-clauses on n Boolean variables and their negations. As m and n tend to infinity in the ratio m/n --> alpha, the problem is known to have a phase transition at alpha_c = 1, below which the probability that the formula is satisfiable tends to one and above which it tends to zero. We determine the finite-size scaling about this transition, namely the scaling of the maximal window W(n,delta) = (alpha_-(n,delta),alpha_+(n,delta)) such that the probability of satisfiability is greater than 1-delta for alpha < alpha_- and is less than delta for alpha > alpha_+. We show that W(n,delta)=(1-Theta(n^{-1/3}),1+Theta(n^{-1/3})), where the constants implicit in Theta depend on delta. We also determine the rates at which the probability of satisfiability approaches one and zero at the boundaries of the window. Namely, for m=(1+epsilon)n, where epsilon may depend on n as long as |epsilon| is sufficiently small and |epsilon|*n^(1/3) is sufficiently large, we show that the probability of satisfiability decays like exp(-Theta(n*epsilon^3)) above the window, and goes to one like 1-Theta(1/(n*|epsilon|^3)) below the window. We prove these results by defining an order parameter for the transition and establishing its scaling behavior in n both inside and outside the window. Using this order parameter, we prove that the 2-SAT phase transition is continuous with an order parameter critical exponent of 1. We also determine the values of two other critical exponents, showing that the exponents of 2-SAT are identical to those of the random graph.Comment: 57 pages. This version updates some reference

Sampling Techniques for Boolean Satisfiability

Boolean satisfiability ({\SAT}) has played a key role in diverse areas spanning testing, formal verification, planning, optimization, inferencing and the like. Apart from the classical problem of checking boolean satisfiability, the problems of generating satisfying uniformly at random, and of counting the total number of satisfying assignments have also attracted significant theoretical and practical interest over the years. Prior work offered heuristic approaches with very weak or no guarantee of performance, and theoretical approaches with proven guarantees, but poor performance in practice. We propose a novel approach based on limited-independence hashing that allows us to design algorithms for both problems, with strong theoretical guarantees and scalability extending to thousands of variables. Based on this approach, we present two practical algorithms, {\UniformWitness}: a near uniform generator and {\approxMC}: the first scalable approximate model counter, along with reference implementations. Our algorithms work by issuing polynomial calls to {\SAT} solver. We demonstrate scalability of our algorithms over a large set of benchmarks arising from different application domains.Comment: MS Thesis submitted to Rice Universit

Publishing Microdata with a Robust Privacy Guarantee

Today, the publication of microdata poses a privacy threat. Vast research has striven to define the privacy condition that microdata should satisfy before it is released, and devise algorithms to anonymize the data so as to achieve this condition. Yet, no method proposed to date explicitly bounds the percentage of information an adversary gains after seeing the published data for each sensitive value therein. This paper introduces beta-likeness, an appropriately robust privacy model for microdata anonymization, along with two anonymization schemes designed therefor, the one based on generalization, and the other based on perturbation. Our model postulates that an adversary's confidence on the likelihood of a certain sensitive-attribute (SA) value should not increase, in relative difference terms, by more than a predefined threshold. Our techniques aim to satisfy a given beta threshold with little information loss. We experimentally demonstrate that (i) our model provides an effective privacy guarantee in a way that predecessor models cannot, (ii) our generalization scheme is more effective and efficient in its task than methods adapting algorithms for the k-anonymity model, and (iii) our perturbation method outperforms a baseline approach. Moreover, we discuss in detail the resistance of our model and methods to attacks proposed in previous research.Comment: VLDB201